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arXiv:1806.03940 · Journal of Mathematical Physics

Symmetries of the Dirac quantum walk and emergence of the de Sitter group

Luca Apadula, Alessandro Bisio, Giacomo Mauro D’Ariano, Paolo Perinotti

Published Aug 1, 2020• 4 citations
arXivHTMLPDF

Abstract

A quantum walk describes the discrete unitary evolution of a quantum particle on a discrete graph. Some quantum walks, referred to as the Weyl and Dirac walks, provide a description of the free evolution of relativistic quantum fields in the small wave-vector regime. The clash between the intrinsic discreteness of quantum walks and the continuous symmetries of special relativity is resolved by giving a definition of change of inertial frame in terms of a change of values of the constants of motion, which leaves the walk operator unchanged. Starting from the family of 1 + 1 dimensional Dirac walks with all possible values of the mass parameter, we introduce a unique walk encompassing the latter as an extra degree of freedom, and we derive its group of changes of inertial frames. This symmetry group contains a non-linear realization of SO+(2,1)⋉R3; since one of the two space-like dimensions does not correspond to an actual spatial degree of freedom but rather the mass, we interpret it as a 2 + 1 dimensional de Sitter group. This group also contains a non-linear realization of the proper orthochronous Poincaré group SO+(1,1)⋉R2 in 1 + 1 dimension, as the ones considered within the framework of doubly special relativity, which recovers the usual relativistic symmetry in the limit of small wave-vectors and masses. Surprisingly, for the Dirac walk with a fixed value of the mass parameter, the group of allowed changes of reference frame does not have a consistent interpretation in the limit of small wave-vectors.

DOI
10.1063/1.5144731

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A quantum walk describes the discrete unitary evolution of a quantum particle on a discrete graph. Some quantum walks, referred to as the Weyl and Dirac walks, provide a description of the free evolution of relativistic quantum fields in the small wave-vector regime. The clash between the intrinsic discreteness of quantum walks and the continuous symmetries of special relativity is resolved by giving a definition of change of inertial frame in terms of a change of values of the constants of motion, which leaves the walk operator unchanged. Starting from the family of 1 + 1 dimensional Dirac walks with all possible values of the mass parameter, we introduce a unique walk encompassing the latter as an extra degree of freedom, and we derive its group of changes of inertial frames. This symmetry group contains a non-linear realization of SO+(2,1)⋉R3; since one of the two space-like dimensions does not correspond to an actual spatial degree of freedom but rather the mass, we interpret it as a 2 + 1 dimensional de Sitter group. This group also contains a non-linear realization of the proper orthochronous Poincaré group SO+(1,1)⋉R2 in 1 + 1 dimension, as the ones considered within the framework of doubly special relativity, which recovers the usual relativistic symmetry in the limit of small wave-vectors and masses. Surprisingly, for the Dirac walk with a fixed value of the mass parameter, the group of allowed changes of reference frame does not have a consistent interpretation in the limit of small wave-vectors.
In this paper we consider quantum walks whose evolution converges to the Dirac equation in the limit of small wave-vectors. We show exact Fast Fourier implementation of the Dirac quantum walks in one, two, and three space dimensions. The behaviour of particle states—defined as states smoothly peaked in some wave-vector eigenstate of the walk—is described by an approximated dispersive differential equation that for small wave-vectors gives the usual Dirac particle and antiparticle kinematics. The accuracy of the approximation is provided in terms of a lower bound on the fidelity between the exactly evolved state and the approximated one. The jittering of the position operator expectation value for states having both a particle and an antiparticle component is analytically derived and observed in the numerical implementations.
A discrete-time Quantum Walk (QW) is essentially an operator driving the evolution of a single particle on the lattice, through local unitaries. Some QWs admit a continuum limit, leading to familiar PDEs (e.g. the Dirac equation). Recently it was discovered that prior grouping and encoding allows for more general continuum limit equations (e.g. the Dirac equation in (1+ 1) curved spacetime). In this paper, we extend these results to arbitrary space dimension and internal degree of freedom. We recover an entire class of PDEs encompassing the massive Dirac equation in (3 + 1) curved spacetime. This means that the metric field can be represented by a field of local unitaries over a lattice.
Spin of elementary particles is the only kinematic degree of freedom not having classical corre- spondence. It arises when seeking for the finite-dimensional representations of the Lorentz group, which is the only symmetry group of relativistic quantum field theory acting on multiple-component quantum fields non-unitarily. We study linear transformations, acting on the space of spatial and proper-time velocities rather than on coordinates. While ensuring the relativistic in- variance, they avoid these two exceptions: they describe the spin degree of freedom of a pointlike particle yet at a classical level and form a compact group hence with unitary finite-dimensional rep- resentations. Within this approach changes of the velocity modulus and direction can be accounted for by rotations of two independent unit vectors. Dirac spinors just provide the quantum description of these rotations.
We consider the Weyl quantum walk in 3+1 dimensions, that is a discrete-time walk describing a particle with two internal degrees of freedom moving on a Cayley graph of the group [Formula: see text], which in an appropriate regime evolves according to Weyl's equation. The Weyl quantum walk was recently derived as the unique unitary evolution on a Cayley graph of [Formula: see text] that is homogeneous and isotropic. The general solution of the quantum walk evolution is provided here in the position representation, by the analytical expression of the propagator, i.e. transition amplitude from a node of the graph to another node in a finite number of steps. The quantum nature of the walk manifests itself in the interference of the paths on the graph joining the given nodes. The solution is based on the binary encoding of the admissible paths on the graph and on the semigroup structure of the walk transition matrices.This article is part of the themed issue 'Second quantum revolution: foundational questions'.
We introduce a new class of p-adic Dirac equations, formulated in the standard axiomatic framework of quantum mechanics, in which the ordinary momentum operators are replaced by bounded non-local operators built from arbitrary integrable radial kernels. We diagonalize the free Dirac Hamiltonian in momentum space, construct its plane-wave solutions, and determine its spectrum, which is pure point, with eigenvalues of infinite multiplicity; in the massless case we describe the zero-energy subspace explicitly. We identify the charge-conjugation symmetry of the model: it is given by an antiunitary operator whose square is minus the identity, it exchanges the positive- and negative-energy subspaces, and in an external electromagnetic field it reverses the sign of both the charge and the vector potential. We then discretize the free equation in two ways, both giving genuine continuous-time quantum walks rather than discrete-time, coined walks: one on a countable graph and one on a finite, tree-like graph. In both cases the discretization is exact: the walk is the restriction of the p-adic Dirac evolution to an invariant subspace, and its Hamiltonian is a Dirac-type combination of weighted graph Laplacians with ultrametric weights. We prove that the Born-rule transition probabilities, summed over the internal degrees of freedom and averaged over the initial internal state, form a doubly stochastic matrix at every instant, and that, as for any non-trivial continuous-time quantum walk, this family of matrices is not a Markov semigroup. We discuss how the construction can serve as the foundation of a quantum network with four-component spinor degrees of freedom, complementing earlier, non-relativistic p-adic quantum neural networks, and prove global well-posedness of the associated nonlinear network equation. We close with a discussion of open problems.
This paper derives the Dirac equation as the logically forced relativistic completion of the de Broglie wave dynamics established in prior work of this series. Starting from the logical foundation that yields Lorentz/Poincaré invariance, we promote the non-relativistic framework (continuous spacetime, Fourier duality, local plane-wave structures, and probability conservation via statistical energy flux) to the relativistic regime. We show that the requirements of exact local relativistic plane-wave structures (with phase gradients satisfying the mass-shell condition pμpμ=m2c2p^\mu p_\mu = m^2 c^2pμpμ​=m2c2), first-order time evolution (needed for clean probability conservation), and full Lorentz covariance together force the introduction of a specific non-commuting algebraic structure and a multi-component wave function. The Dirac equation emerges as the unique linear differential operator of order at most one in the spacetime derivatives that consistently glues these local structures across smooth backgrounds while preserving a conserved four-current with positive-definite time component. This multi-component structure arises naturally as the minimal algebraic consequence of the constraints, without any separate postulate for intrinsic angular momentum. In exact analogy with the emergence of the finite invariant causal speed in special relativity, the Dirac dynamics is revealed not as an ad-hoc linearization of the Klein--Gordon equation, but as the mathematically inevitable description of indivisible particles in continuous relativistic spacetime. The derivation thereby completes the single-particle sector of the series, demonstrating that core structures of relativistic quantum mechanics flow deductively from a small number of deep, nearly tautological commitments.